Perfectoid universal covers of curves

In a previous post I mumbled a bit about perfectoid covering spaces of rigid analytic varieties.  In this post, I want to sketch a fun special case.

Fix ${K/\mathbf{Q}_p}$ an algebraically closed nonarchimedean field. Recall the following from Scholze-Weinstein: given a cofiltered inverse system ${(X_i)_{i\in I}}$ of rigid spaces over ${K}$ with qcqs transition maps, if there is a perfectoid space ${X_\infty}$ such that ${X_\infty \sim \lim_{\leftarrow i} X_i}$, then ${X_\infty}$ is unique up to unique isomorphism. In this scenario, I will often write ${X_\infty = \lim_{\leftarrow i\in I} X_i}$ when such a perfectoid space exists.

Now, fix a smooth proper curve ${\mathfrak{X}}$ over ${\mathrm{Spec}(K)}$ of genus ${g\geq 1}$, with associated adic space ${X= \mathfrak{X}^{\mathrm{ad}}}$ over ${\mathrm{Spa}(K)}$.

Lemma. There is a natural equivalence ${\mathrm{F\acute{E}t}(\mathfrak{X}) \cong \mathrm{F\acute{E}t}(X)}$.

Proof: This follows from Lutkebohmert’s ${p}$-adic Riemann existence theorem, or from a very cool theorem of Fresnel and Matignon: any irreducible, quasicompact, separated one-dimensional rigid analytic space is either affinoid or projective. $\Box$

Now fix a base point ${x=\mathrm{Spa}(K) \in X}$, and let ${\pi_1(X,x)=\pi_1(\mathfrak{X},x)}$ denote the usual étale fundamental group from SGA; this is a profinite group depending only on ${g}$ (up to isomorphism), and there is a natural fiber functor

$\displaystyle F_x : \mathrm{F\acute{E}t}(X) \cong \pi_1(X,x)-\mathrm{Sets},$

where ${G}$-Sets (for ${G}$ a profinite group) denotes the category of finite sets with continuous (left) ${G}$-action. For any open subgroup ${U < \pi_1(X,x)}$, we have a corresponding connected finite étale cover ${X_U \rightarrow X}$ (which is uniquely algebraizable to a covering ${\mathfrak{X}_U \rightarrow \mathfrak{X}}$).

Theorem. Suppose the Néron model of ${\mathfrak{A}=\mathrm{Jac}(\mathfrak{X})}$ is an abelian scheme over ${\mathrm{Spec}(\mathcal{O}_K)}$. Then there is a natural perfectoid space ${\tilde{X}}$ such that

$\displaystyle \tilde{X} = \lim_{\substack{\leftarrow \\ U\rightarrow {1}}} X_U.$

Under the natural map ${q: \tilde{X} \rightarrow X}$, ${\tilde{X}}$ is naturally a ${\pi_1(X,x)}$-torsor over ${X}$, and any ${Y \in \mathrm{F\acute{E}t}(X)}$ can be recovered from ${F_x(Y)}$ as the categorical quotient ${\tilde{X} \times_{\pi_1(X,x)}F_x(Y)}$.

In the remainder of this post, I’ll sketch the construction of ${\tilde{X}}$. Here’s the idea: Let ${A = \mathfrak{A}^{ad}}$ be the rigid analytic Jacobian of ${X}$, so our point ${x}$ gives a natural closed immersion ${X \rightarrow A}$. On fundamental groups we get a natural surjection

$\displaystyle \alpha: \pi_1(X,x) \twoheadrightarrow \pi_1(X,x)^{\mathrm{ab}} \cong \pi_1(A,e) \simeq \hat{\mathbf{Z}}^{2g}.$

Set ${U_n = \alpha^{-1}(p^n \hat{\mathbf{Z}}^{2g})}$, and let ${X_n := X_{U_n}}$ denote the associated covering of ${X}$; these form an inverse system in the obvious way. We are going to first construct a perfectoid space ${X_\infty}$ such that ${X_\infty = \lim_{\leftarrow n} X_n}$, and then carefully pile all the remaining ${X_U}$‘s on top of ${X_\infty}$, using the almost purity theorem to ensure that this pileup remains perfectoid.

Let ${A_n \in \mathrm{F\acute{E}t}(A)}$ denote a copy of ${A}$ mapping to ${A}$ under ${[p^n]}$, so ${A_n \rightarrow A}$ is a finite étale covering of degree ${p^{2gn}}$. These form an inverse system in the obvious way.

Lemma. The map ${X_n \rightarrow X}$ is obtained from the embedding ${X \rightarrow A}$ by pulling back under ${A_n \rightarrow A}$.

Proof: Easy exercise. $\Box$

Lemma. There is a perfectoid space ${A_\infty}$ such that ${A_\infty = \lim_{\leftarrow n} A_n}$.

Proof: This is Lemme A.16 in Pilloni-Stroh’s “Cohomologie coherent et representations Galoisiennes”. $\Box$

Lemma (“The pullback lemma”). Let ${f:X \rightarrow Y}$ be a morphism of rigid analytic spaces, and let ${Z \rightarrow Y}$ be a perfectoid space. If ${f}$ is “blah”, where “blah” ${\in}$ ${\{}$open immersion, closed immersion, finite étale, étale${\}}$, then ${Z \times_{Y} X}$ exists as a perfectoid space and ${Z \times_{Y} X \rightarrow Z}$ is “blah”.

If ${Z \sim \lim_{\leftarrow i} Y_i}$ for some cofiltered inverse system of rigid spaces ${Y_i /Y}$ with qcqs transition maps, then ${Z \times_{Y} X \sim \lim_{\leftarrow i}(Y_i \times_{Y}X)}$.

Proof: It suffices to work locally on ${X}$ and ${Y}$. We give a brief sketch:
Open immersion: Immediate.
Closed immersion: Pull back the ideal sheaf defining ${X}$ in ${Y}$ and argue as in § II.2 of Scholze’s torsion paper.
Finite étale: Immediate from the almost purity theorem.
Etale: Any such morphism is locally a composite of finite étale morphisms and open immersions, so we reduce to those cases. $\Box$

Lemma. The fiber product ${X_\infty = A_\infty \times_{A} X}$ exists as a perfectoid space, and ${X_\infty \sim \lim_{\leftarrow n} X_n}$.

Proof: Immediate from the pullback lemma applied to the case where ${Y=A}$, ${(Y_i)=(A_n)}$, ${Z=A_{\infty}}$. $\Box$

Lemma. For any open subgroup ${H < \pi_1(X,x)}$, there is a natural perfectoid space ${X_{H,\infty}}$ such that ${X_{H,\infty} = \lim_{\leftarrow n}X_{H \cap U_n}}$, and ${X_{H,\infty}}$ is finite étale over ${X_\infty}$.

Proof: The space ${X_{H,\infty}}$ is naturally a connected component of ${X_{H} \times_{X} X_\infty}$; by the pullback lemma, this fiber product is perfectoid and finite étale over ${X_\infty}$. The inclusion map ${X_{H,\infty} \rightarrow X_{H} \times_{X} X_\infty}$ is finite étale, so we deduce that ${X_{H,\infty} \rightarrow X_\infty}$ is finite étale. $\Box$

The spaces ${(X_{H,\infty})_{H}}$ form a cofiltered inverse system of perfectoid spaces over ${X_\infty}$, with finite étale transition maps.

Lemma. The inverse limit

$\displaystyle \lim_{\substack{\leftarrow \\ H \rightarrow {1}}} X_{H,\infty}$

exists as a perfectoid space.

Proof: Argue locally on ${X_\infty}$: let ${V \subset X_\infty}$ be an open affinoid perfectoid subset, with preimage ${V_H \subset X_{H,\infty}}$. By the definition of finite étale morphisms, each ${V_H}$ is affinoid perfectoid; since the category of affinoid perfectoid spaces admits all limits, we get ${\lim_{\leftarrow H} V_H}$ as an affinoid perfectoid space over ${V}$. $\Box$

Final lemma. Let ${\tilde{X}}$ be the perfectoid space constructed in the previous lemma. Then ${\tilde{X} \sim \lim_{U \rightarrow 1} X_U}$.

Proof: Easy consequence of all the above. $\Box$